Penalized likelihood estimation of a fixed-effect and a mixed-effect transfer function model

by Hansen, Elizabeth Ann.

Abstract (Summary)

Motivated by the need of estimating the main spawning period of North Sea
cod, we develop a common transfer function model with a panel of contemporaneously
correlated times series data. This model incorporates (i) the smoothness on the
parameters by assuming that the second differences are small and (ii) the contemporaneous
correlation by assuming that the errors have a general variance-covariance
matrix. Penalized likelihood estimation of this model requires an iterative procedure
that is developed in this work. We develop three methods for determining confidence
bands: frequentist, Bayesian, and bootstrap (both nonparametric and parametric).
A simulation study on the frequentist and Bayesian confidence bands motivated by
the cod spawning data is conducted and the results of those simulations are compared.
The model is then used on the cod spawning data, with all confidence bands
computed. The results of this analysis are discussed. We then delve further into our
model by discussing the theory behind this model. We prove a theorem that shows
that the estimated regression parameter vector is a consistent estimate of the true
regression parameter. We further prove that this estimated regression parameter vector
has an asymptotic normal distribution. Both theorems are proved while assuming
mild conditions.
We further develop our model by incorporating between-series variation in the
transfer function, with the random effect assumed to have a normal distribution with a
smooth” mean vector. We implement the EM algorithm to do the penalized likelihood
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estimation. We consider five different specifications of the variance-covariance matrix
of the random transfer function model, namely, a general variance-covariance matrix,
a diagonal matrix, a multiple of the identity matrix, an autoregressive matrix of order
one, and a multiplicative error specification. Since the computation of confidence
bands would lead to numerical problems, we introduce a bootstrap approach for
estimating the confidence bands. We consider both the nonparametric and parametric
bootstrap approaches. We then apply this model to estimate the cod spawning period,
while also looking into the different specifications of the variance-covariance matrix
of the random effect, the two types of bootstrapped confidence bands, and model
checking.
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